  
  [1X4 [33X[0;0YAdvanced functions for Lie [22Xp[122X[101X[1X-rings[133X[101X
  
  [33X[0;0YThis  chapter  describes a few more advanced functions available for generic
  Lie [22Xp[122X-rings.[133X
  
  
  [1X4.1 [33X[0;0YSchur multipliers[133X[101X
  
  [33X[0;0YThe  package contains a method to determine the Schur multipliers of the Lie
  [22Xp[122X-rings in the family defined by a generic Lie [22Xp[122X-ring.[133X
  
  [1X4.1-1 LiePSchurMult[101X
  
  [33X[1;0Y[29X[2XLiePSchurMult[102X( [3XL[103X ) [32X function[133X
  
  [33X[0;0YThe  function takes as input a generic Lie [22Xp[122X-ring [3XL[103X and determines a list of
  possible  Schur  multipliers,  each described by its abelian invariants, for
  the  Lie [22Xp[122X-rings in the family described by [3XL[103X. For each entry in the list of
  Schur  multipliers there is a description of those parameters which give the
  considered  entry.  This  description consists of two lists [10Xunits[110X and [10Xzeros[110X.
  Both  consist  of  rational functions over the parameters of the Lie [22Xp[122X-ring.
  The parameters described by these lists are those which evaluate to zero for
  each  rational  function  in  [10Xzeros[110X  and  do  not  evaluate to zero for each
  rational function in [10Xunits[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XLL := LiePRingsByLibrary(7);;[127X[104X
    [4X[25Xgap>[125X [27XL := Filtered(LL, x -> Length(ParametersOfLiePRing(x))=2)[1]; [127X[104X
    [4X[28X<LiePRing of dimension 7 over prime p with parameters [ x, y ]>[128X[104X
    [4X[25Xgap>[125X [27XNumberOfLiePRingsInFamily(L);[127X[104X
    [4X[28Xp^2-p[128X[104X
    [4X[25Xgap>[125X [27XRingInvariants(L);[127X[104X
    [4X[28Xrec( units := [ x ], zeros := [  ] )[128X[104X
    [4X[25Xgap>[125X [27Xss := LiePSchurMult(L);[127X[104X
    [4X[28X[ rec( norm := [ p ], units := [ x, y ], zeros := [ x*y^2-x*y+1 ] ), [128X[104X
    [4X[28X  rec( norm := [ p^2 ], units := [ x ], zeros := [ x*y ] ), [128X[104X
    [4X[28X  rec( norm := [ p ], units := [ x, x*y^2-x*y+1, y ], zeros := [  ] ) ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YIn  this example, [3XL[103X defines a generic Lie [22Xp[122X-ring with two parameters and the
  [10XRingInvariants[110X  of  [3XL[103X  show  that  the  parameter  [22Xx[122X should be non-zero. The
  function   [10XLiePSchurMult(L)[110X   yields  that  there  are  two  possible  Schur
  multipliers  for  the  Lie  [22Xp[122X-rings  in  the family defined by [3XL[103X: the cyclic
  groups  of order [22Xp[122X and of order [22Xp^2[122X. The second option only arises if [22Xxy = 0[122X
  and thus, as [22Xx[122X is non-zero, if [22Xy = 0[122X.[133X
  
  [33X[0;0YThe  package also contains a function that tries to determine the numbers of
  values  of  the  parameters  satisfying the conditions of a description of a
  Schur  multiplier. This succeeds in many cases and returns a polynomial in [22Xp[122X
  in this case. If it does not succeed then it returns [9Xfail[109X.[133X
  
  [1X4.1-2 ElementNumbers[101X
  
  [33X[1;0Y[29X[2XElementNumbers[102X( [3Xpp[103X, [3Xss[103X ) [32X function[133X
  
  [33X[0;0YWe continue the above example.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XElementNumbers(ParametersOfLiePRing(L), ss);[127X[104X
    [4X[28Xrec( norms := [ [ p^2 ], [ p ] ], numbs := [ p-1, p^2-2*p+1 ] )[128X[104X
  [4X[32X[104X
  
  
  [1X4.2 [33X[0;0YAutomorphism groups[133X[101X
  
  [33X[0;0YThe  package  contains  a  function  that  determines  a description for the
  automorphism  groups  of  the Lie [22Xp[122X-rings in the family defined by a generic
  Lie [22Xp[122X-ring.[133X
  
  [1X4.2-1 AutGroupDescription[101X
  
  [33X[1;0Y[29X[2XAutGroupDescription[102X( [3XL[103X ) [32X function[133X
  
  [33X[0;0YEach automorphism of [3XL[103X is defined by its images on a generating set of [3XL[103X. If
  [22Xl_1,  ...,  l_n[122X  is a basis of [3XL[103X and [22Xl_1, ..., l_d[122X is a generating set, then
  each  automorphism  is defined by the images of [22Xl_1, ..., l_d[122X and each image
  is  an  integral linear combination of the basis elements [22Xl_1, ..., l_n[122X. The
  function  [10XAutGroupDescription[110X  returns  a matrix containing a description of
  the  coefficients  in  each linear combination and a list of relations among
  these coefficients. We consider two examples.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL := Filtered(LL, x -> Length(ParametersOfLiePRing(x))=2)[1];[127X[104X
    [4X[28X<LiePRing of dimension 7 over prime p with parameters [ x, y ]>[128X[104X
    [4X[25Xgap>[125X [27XAutGroupDescription(L);[127X[104X
    [4X[28Xrec( auto := [ [ 1, 0, A13, A14, A15, A16, A17 ], [128X[104X
    [4X[28X               [ 0, 1, A23, A24, A25, A26, A27 ] ], [128X[104X
    [4X[28X     eqns := [ [  ], [  ] ] )[128X[104X
    [4X[25Xgap>[125X [27XL := Filtered(LL, x -> Length(ParametersOfLiePRing(x))=2)[2];[127X[104X
    [4X[28X<LiePRing of dimension 7 over prime p with parameters [ x, y ]>[128X[104X
    [4X[25Xgap>[125X [27XAutGroupDescription(L);[127X[104X
    [4X[28Xrec( auto := [ [ A22^3, 0, A13, A14, A15, A16, A17 ], [128X[104X
    [4X[28X               [ 0, A22, A23, A24, A25, A26, A27 ] ], [128X[104X
    [4X[28X     eqns := [ [ A22*A24-1/2*A23^2, A22^2*y-y, [128X[104X
    [4X[28X                 A22*A23^2*y-2*A24*y, A22^4-1, [128X[104X
    [4X[28X                 A23^4*y-4*A24^2*y, A22^3*A23^2-2*A24, [128X[104X
    [4X[28X                 A22^2*A23^4-4*A24^2, A22*A23^6-8*A24^3, [128X[104X
    [4X[28X                 A23^8-16*A24^4 ] ] )[128X[104X
  [4X[32X[104X
  
  [33X[0;0YIn  both  cases,  [3XL[103X  is  generated by the first two entries in its basis and
  hence  the  automorphism group matrix has two rows and seven columns. In the
  first   case,  [3XL[103X  has  [22Xp^10[122X  automorphisms  inducing  the  identity  on  the
  Frattini-quotient  of  [3XL[103X.  In the second case, the automorphism group matrix
  shows  that  each  automorphism induces a certain type of diagonal matrix on
  the  Frattini-quotient  of  [3XL[103X  and  there  are  further  equations among the
  coefficients  of the matrix. These further equations are equivalent to [22XA22^2
  = 1[122X and [22XA24 = A22 A23^2 / 2[122X. Hence [3XL[103X has [22X2 p^9[122X automorphisms.[133X
  
  [33X[0;0YThe  entry  [10Xeqns[110X is a list of lists. The equations in the [22Xi[122X-th entry of this
  list have to be satisfied mod [22Xp^i[122X.[133X
  
  [33X[0;0YIn   a   few  special  cases,  the  function  returns  a  list  of  possible
  automorphisms  together with related equations and conditions. We exhibit an
  example.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL := LiePRingsByLibrary(7)[489];[127X[104X
    [4X[28X<LiePRing of dimension 7 over prime p with parameters [ x ]>[128X[104X
    [4X[25Xgap>[125X [27XAutGroupDescription(L);[127X[104X
    [4X[28X[ rec( auto := [ [ 1, 0, A13, A14, A15, A16, A17 ], [128X[104X
    [4X[28X                 [ 0, 1, A23, A24, A25, A26, A27 ] ], [128X[104X
    [4X[28X       comment := "p^8 automorphisms",[128X[104X
    [4X[28X       eqns := [ [ A13^2*x-A13*A23+2*A15*x+A14-A25, [128X[104X
    [4X[28X              -A13*A23*x+A14*x+A23^2-A25*x-2*A24 ] ] ), [128X[104X
    [4X[28X  rec( auto := [ [ 0, A12, A13, A14, A15, A16, A17 ], [128X[104X
    [4X[28X                 [ -x, 0, A23, A24, A25, A26, A27 ] ], [128X[104X
    [4X[28X      comment := "p^8 automorphisms when x <> 0 mod p", [128X[104X
    [4X[28X      eqns := [ [ A12^2*A24*x-A12*A13*A23*x+A12*A13*x^2[128X[104X
    [4X[28X                  +2*A12*A15*x^2+A12*A14*x-A13^2*x+A13*x+A15*x-A14, [128X[104X
    [4X[28X                  -A12^2*A23*x^3+A12*A13*x^3+A12*A23^2*x-A12*A25*x^2[128X[104X
    [4X[28X                  -2*A12*A24*x+A13*A23*x+A13*x^2-A15*x^2+A23*x+A25*x-A24 ], [128X[104X
    [4X[28X                [ A12*x+1 ] ] ) ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YIn  this  example  [22XA12 x = -1[122X modulo [22Xp^2[122X. We note that different choices for
  [22XA12[122X  do not give different automorphisms. Hence a single solution for [22XA12[122X is
  sufficient to describe all automorphisms.[133X
  
  [33X[0;0Y [133X
  
